Let н be a complex, separable, infinite dimensional Hilbert space, T ε（Ｈ）. （ｕ＋Ｋ）（T） denotes the （ｕ＋ｋ）-orbit of T, i.e., （ｕ＋ｋ）（T） = {R-1TR: R is invertible and of the form unitary plus compact}. Let be an analytic and simply connected Cauchy domain in C and n ε N. A（, n） denotes the class of operators, each of which satisfies （i） T is essentially normal; （ii） σ（T） =, ρF（T） ∩ σ（T） = ; （iii） ind （λ-T） = -n, nul （λ-T） = 0 （λ∈Ω ）. It is proved that given T1, T2 ε A（, n） and ε > 0, there exists a compact operator K with K <ε such that T1 +Kε （ｕ＋ｋ）（T2）. This result generalizes a result of P. S. Guinand and L. Marcoux [6,15]. Furthermore, the authors give a character of the norm closure of （ｕ＋Ｋ）（T）, and prove that for each T ε A（, n）, there exists a compact （SI） perturbation of T whose norm can be arbitrarily small.
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